A328768 The first primorial based variant of arithmetic derivative: a(prime(i)) = A002110(i-1), where prime(i) = A000040(i), a(u*v) = a(u)*v + u*a(v), with a(0) = a(1) = 0.

0, 0, 1, 2, 4, 6, 7, 30, 12, 12, 17, 210, 20, 2310, 67, 28, 32, 30030, 33, 510510, 44, 104, 431, 9699690, 52, 60, 4633, 54, 148, 223092870, 71, 6469693230, 80, 652, 60077, 192, 84, 200560490130, 1021039, 6956, 108, 7420738134810, 229, 304250263527210, 884, 114, 19399403, 13082761331670030, 128, 420, 145, 90124, 9292, 614889782588491410, 135, 1116, 324

Offset

0,4

Links

Antti Karttunen, Table of n, a(n) for n = 0..1001

Index entries for sequences related to primorial numbers

Formula

a(n) = n * Sum e_j * A276085(p_j)/p_j for n = Product p_j^e_j, where for primes p, A276085(p) = A002110(A000720(p)-1).

a(n) = n * Sum e_j * (p_j)#/(p_j^2) for n = Product p_j^e_j with (p_j)# = A034386(p_j).

For all n >= 0, A276150(a(n)) = A328771(n).

Prog

(PARI)
A002110(n) = prod(i=1, n, prime(i));
A328768(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]*A002110(primepi(f[i, 1])-1)/f[i, 1]));
(PARI) A328768(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]*A002110(primepi(f[i, 1]))/(f[i, 1]^2)));

Crossrefs

Cf. A002110, A108951, A276085, A276150, A328771.

Cf. also A003415, A258851, A328769, A328845, A328846.

Sequence in context: A105176 A061937 A065804 * A275693 A135033 A317647

Adjacent sequences: A328765 A328766 A328767 * A328769 A328770 A328771

Keyword

nonn

Author

Antti Karttunen, Oct 28 2019

Status

approved